Bayesian Equalization for LDPC Channel Decoding

Abstract

Abstract-We describe the channel equalization problem and its prior estimate of the channel estate information (CSI), as a joint Bayesian estimation problem to improve each symbol posterior estimates at the input of the channel decoder. Our solution takes into consideration not only the uncertainty due to the noise in the channel, but also the uncertainty in the CSI estimate. The marginalization to compute each posterior from the full posterior cannot be computed in linear time, because it depends on all the transmitted symbols. Hence, we also put forward an approximate posterior, inspired by the BCJR algorithm, which is optimal from the Kullback-Leibler divergence viewpoint and presents a complexity identical to the BCJR algorithm. We also use a graphical model representation of the full posterior, in which the proposed approximation can be readily understood. The proposed posterior estimates are more accurate than those computed using the ML estimate for the CSI. To illustrate this point, we measure the bit error rate at the output of a Low-Density Parity-Check decoder, which needs the exact posterior for each symbol to detect the incoming word and it is sensitive to a mismatch in those posterior estimates.

Turbo EP-Based Equalization: A Filter-Type Implementation

“…where q rℓs E pu k q is an extrinsic marginal distribution computed as 2 q rℓs E pu k q " q rℓs pu k q{p rℓs…”

“…The equalizer produces a probabilistic estimation of the transmitted data given the vector of observations [1]. Significant improvements are found when the previous estimation is given to a probabilistic channel decoder [2]. Equalization can be done in the frequency domain to avoid complexity problems associated with the inverse of covariance matrices [3].…”

Joint Detection/Decoding Algorithms for Nonbinary LDPC Codes over ISI Channels

“…The inter-symbol interference (ISI) is a common phenomenon in high-density digital recording systems and wireless communication systems [13]. Different equalizers have been proposed in the literature [14][15][16][17][18][19][20][21][22][23]. Since the invention of the turbo codes [24], the rediscovery of the LDPC codes [1], and most importantly, the success of the applications of turbo principle to equalizations [19] [25][26], many works have been done to apply turbo principles to coded ISI channels [27][28][29][30][31][32][33], where binary convolutional codes, turbo codes or LDPC codes are usually considered as the "outer codes" of the serial concatenated system.…”